Measure theory via locales

Abstract: There are two oftentimes unspoken truths in measure theory. 1) Practically all useful measures in practice are given by Radon measures. 2) One does not really care so much about the sigma-algebra of measurable sets, but rather about its quotient by the ideal of null sets.

The quotient of measurable sets by null sets is, in the case of a given Radon measure, an example of what is called a measurable locale, and can be treated like a (usually point-free) space. We argue that this measurable locale can be constructed directly from a Grothendieck topology on the poset of compact sets. This opens the door to a purely sheaf-theoretic perspective on measure theory. As an application, we show that the locale of sublocales of a given Hausdorff space X equipped with a Radon measure can be equipped with a natural extension of the measure, invariant under measure preserving homeomorphisms.

mathematical physicsalgebraic topologycategory theory

Audience: researchers in the topic

( paper )

Topology and Geometry Seminar (Texas, Kansas)